PID Controls of Motors

Howie Choset(thanks to Gary Fedder)

http://www.library.cmu.edu/ctms/ctms/examples/motor/motor.htm

Controls• Review of Motor Model • Open Loop (controller-free) Response• Proportional Control

– Stable– Faster Response = Bigger Overshoot– Steady State Error

• PI Control– Maintain Stability– Decrease Steady State Error = Bigger Overshoot

• PID Control– Derivative term reduces overshoot, settling time

• Feed Forward– Overcome damping

Mass-Spring-Damper Model(Analogy)

Model of mass spring damper system

z(t) position, z(t) velocityt0 initial time, z(t0), z(t0) initial position & velocity

Review of Motor Model

•electric inductance (L) = [H](VL = L di/dt)

* input (V): Source Voltage* output (theta): position of shaft* The rotor and shaft are assumed to be rigid

• moment of inertia of the rotor (J) [kg.m^2/s^2]* damping ratio of the mechanical system (b) [Nms]* electromotive force constant

•Ke is volt (electromotive force) per radians per second (V/ rad/sec)

• torque constance•Kt is torque amp (Nm/Amp)

* electric resistance (R) = [ohm]

Review of Motor ModelTorque is proportional to current (Lenz’s Law)

Back emf is proportional to motor speed (Faraday’s Law)

Mechanical Equation of Motion

Electrical Equation of Motion KVRidtdiL

KibJ

)(KI )( J ssbss )(KV)(R)I(L ssss

2KRLb JK

V

sss

Assume (K=Ke=Kt)

Solve for sVSpeed, theta dot

Transfer Function of Motor (with Approximations)

=

=

= Open Loop Transfer Function

Can rewrite function in terms of an electrical and mechanical behavior

For small motors, the mechanical behavior dominates (electrical transients die faster).

example motor with equivalent time constants

.

.

.

Electrical time constant on motor is much smaller

Open Loop Response (to a Step)• Apply constant voltage• Slow response time (lag)• Weird Apples-to-Orange relationship between input and output

– If you want to set speed, what voltage do you input?– Weird type of steady state error

• No reaction to perturbations

PlantInputVoltage

OutputSpeed

Closed Loop Controller

Controller EvaluationSteady State ErrorRise Time (to get to ~90%)OvershootSettling Time (Ring) (time to steady state)Stability

Give it a velocity command and get a velocity output

PlantController-

+Ref error voltage

Close the loop analogy

Stability

Asymptotic Stability:

Closed Loop Response (Proportional Feedback)

PlantController-

+R error voltage

Proportional Control

Easy to implementInput/Output units agreeImproved rise time

Steady State Error (true)

P: Rise Time vs. OvershootP: Rise Time vs. Settling time

pK

Voltage = Kp error

Closed Loop Response (PI Feedback)

Plant-

+Ref error voltage

Proportional/Integral Control Ip K

sK 1

No Steady State Error

Bigger Overshoot and SettlingSaturate counters/op-amps

P: Rise Time vs. OvershootP: Rise Time vs. Settling time

I: Steady State Error vs. Overshoot

Voltage = (Kp+1/s Ki) error

Ip Ks

K 1

Closed Loop Response (PID Feedback)

Plant-

+R error voltage

Proportional/Integral/Differential

Quick responseReduced Overshoot

Sensitive to high frequency noiseHard to tune

P: Rise Time vs. OvershootP: Rise Time vs. Settling time

I: Steady State Error vs. Overshoot

D: Overshoot vs. Steady State Error

DIp sKKs

K 1

DIp sKKs

K 1

Voltage = (Kp+1/s Ki + sKd) error

Feed ForwardDecouples Damping from PID

To compute Try different open loop inputs and measure output velocitiesFor each trial i, Tweak from there.

bK

Plant-

+R errorController

++ volt

K

bK

ibb

i

iib KKuK avg ,

.

Volt

Follow a straight line with differential drive

Make both wheels spin the same speedasynchronous – false startwheels can have slight differences (radius, etc)

Make sure both wheels spin the same amount and speedfalse start

More complicated control laws – track orientation m1vref = vref + K1 * thetaerror + K2 * offset error

modeling kinematics of robotdead-reckoning

Error can be difference in wheel velocities or accrued distances

Encoders

Encoders – Incremental

LED Photoemitter

Photodetector

Encoder disk

Encoders - Incremental

Encoders - Incremental

• Quadrature (resolution enhancing)

Where are we?

• If we know our encoder values after the motion, do we know where we are?

Where are we?

• If we know our encoder values after the motion, do we know where we are?

• What about error?

Encoders - Absolute

4 Bit Example

More expensive

Resolution = 360° / 2N

where N is number of tracks